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Robust Optimization — Methodology and Applications1 Robust Optimization — Methodology and Applications1 Aharon Ben-Tal and Arkadi Nemirovski [email protected] [email protected] Faculty of Industrial Engineering and Management Technion – Israel Institute of Technology Technion City, Haifa 32000, Israel Abstract Robust Optimization (RO) is a modeling methodology, combined with computational tools, to process optimization problems in which the data are uncertain and is only known to belong to some uncertainty set. The paper surveys the main results of RO as applied to uncertain linear, conic quadratic and semidefinite programming. For these cases, computa- tionally tractable robust counterparts of uncertain problems are explicitly obtained, or good approximations of these counterparts are proposed, making RO a useful tool for real-world applications. We discuss some of these applications, specifically: antenna design, truss topol- ogy design and stability analysis/synthesis in uncertain dynamic systems. We also describe a case study of 90 LPs from the NETLIB collection. The study reveals that the feasibility properties of the usual solutions of real world LPs can be severely affected by small pertur- bations of the data and that the RO methodology can be successfully used to overcome this phenomenon. AMS 1991 subject classification. Primary: 90C05, 90C25, 90C30. OR/MS subject classification. Primary: Programming/Convex. Key words. Convex optimization, data uncertainty, robustness, linear programming, quadratic program- ming, semidefinite programming, engineering design, Lyapunov stability synthesis. 1 Introduction A generic mathematical programming problem is of the form min x : f (x, ζ) x 0, f (x, ζ) 0, i = 1,...,m (P[ξ]) n 0 0 0 i x0 R,x R { − ≤ ≤ } ∈ ∈ where x is the design vector, the functions f0 (the objective function) and f1,...,fm are structural elements of the problem, and ζ stands for the data specifying a particular problem instance. For real-world optimization problems, the “decision environment” is often characterized by the following facts: F.1. The data are uncertain/inexact; F.2. The optimal solution, even if computed very accurately, may be difficult to implement accurately; F.3. The constraints must remain feasible for all meaningful realizations of the data; F.4. Problems are large-scale (n or/and m are large); F.5. “Bad” optimal solutions (those which become severely infeasible in the face of even relatively small changes in the nominal data) are not uncommon. 1This research was partially supported by the Israeli Ministry of Science grant # 0200-1-98 and the Israel Science Foundation grant # 683/99-10.0 1 F.1 (and in fact F.2 as well) imply that in many cases we deal with uncertain optimization problems – families of the usual (“certain”) optimization problems (P[ζ]) ζ , (1) { | ∈ U} where is some “uncertainty set” in the space of the data. U Fact F.3 implies that a meaningful candidate solution (x0,x) of an uncertain problem (1) is required to satisfy the semi-infinite system of constraints f (x, ζ) x , f (x, ζ) 0 i = 1,...,m (ζ ). (2) 0 ≤ 0 i ≤ ∀ ∈U Fact F.4 on the other hand imposes a severe requirement of being able to process efficiently the semi-infinite system of constraints (2) for large-scale problems. Robust Optimization originating from [3, 4, 5, 11, 12, 13] is a modeling methodology, combined with a suite of computational tools, which is aimed at accomplishing the above requirements. The urgency of having such a methodology stems from the fact F.5, whose validity is well illustrated in the cited papers (and in the examples to follow). In the Robust Optimization methodology, one associates with an uncertain problem (1) its robust counterpart, which is a usual (semi-infinite) optimization program min x0 : f0(x, ζ) x0, fi(x, ζ) 0, i = 1,...,m (ζ ) ; (3) x0,x { ≤ ≤ ∀ ∈U } fesible/optimal solutions of the robust counterpart are called robust feasible/robust optimal solutions of the uncertain problem (1). The major challenges associated with the Robust Optimization methodology are: C.1. When and how can we reformulate (3) as a “computationally tractable” optimization problem, or at least approximate (3) by a tractable problem. C.2. How to specify reasonable uncertainty sets in specific applications. U In what follows, we overview the results and the potential of the Robust Optimization methodology in the most interesting cases of uncertain Linear, Conic Quadratic and Semidefinite Programming. For these cases, the instances of our uncertain optimization problems will be in the conic form min x : cT x x , A x + b K , i = 1,...,m, , (C) n 0 0 i i i x0 R,x R ≤ ∈ ∈ ∈ n o where for every i Ki is mi – either a non-negative orthant R+ (linear constraints), mi 1 mi mi − 2 – or the Lorentz cone L = y R ymi yj (conic quadratic constraints), ( ∈ | ≥ s j=1 ) mi P mi – or a semidefinite cone S+ – the cone of positive semidefinite matrices in the space S of m m symmetric matrices (Linear Matrix Inequality constraints). i × i The class of problems which can be modeled in the form of (C) is extremely wide (see, e.g., [10, 9]). It is also clear what is the structure and what are the data in (C) – the former is the design dimension n, the number of “conic constraints” m and the list of the cones K1,..., Km, while the latter is the collection of matrices and vectors c, A , b m of appropriate sizes. Thus, { i i}i=1 an uncertain problem (C) is a collection min x : x cT x K R , A x + b K , i = 1,...,m (c, A , b m ) n 0 0 0 + i i i i i i=1 x0 R,x R − ∈ ≡ ∈ | { } ∈U ∈ ∈ A0x+b0 (4) | {z } 2 of instances (C) of a common structure (n,m, K ,..., K ) and data (c, A , b m ) varying in 1 m { i i}i=1 a given set . Note that the robust counterpart is a “constraint-wise” notion: in the robust U counterpart of (4), every one of the uncertain constraints A x + b K , i = 0,...,m, of (4) is i i ∈ i replaced with its robust counterpart A x + b K (A , b ) , i i ∈ i ∀ i i ∈Ui where is the projection of the uncertainty set on the space of the data of i-th constraint. Ui U This observation allows us in the rest of the paper to focus on the “pure cases” (all Ki’s are orthants, or all of them are Lorentz cones, or all are the semidefinite cones) only; although the results can be readily applied to the “mixed cases” as well. In the rest of this paper, we consider in turn uncertain Linear, Conic Quadratic and Semidefi- nite programming problems; our major interest is in the relevant version of C.1 in the case when the uncertainty set is an intersection of ellipsoids (which seems to be a general enough model from the viewpoint of C.2). We shall illustrate our general considerations by several examples, mainly coming from engineering (additional examples can be found, e.g., in [2, 3, 7, 11, 12, 13, 14]). We believe that these examples will convince the reader that in many applications taking into ac- count from the very beginning data uncertainty is a necessity which cannot be ignored, and that the Robust Optimization methodology does allow one to deal with this necessity successfully. 2 Robust Linear Programming 2.1 Robust counterpart of uncertain LP In [4] we showed that the robust counterpart min t : t cT x,Ax b (c, A, B) (5) t,x ≥ ≥ ∀ ∈U n o of an uncertain LP T n m n m min c x : Ax b (c, A, b) R R × R (6) x ≥ | ∈U⊂ × × n n o o is equivalent to an explicit computationally tractable problem, provided that the uncertainty set itself is “computationally tractable”. To simplify our presentation, we restrict ourselves U to a restricted family of “computationally tractable” sets, and here is the corresponding result: Theorem 2.1 [[4]] Assume that the uncertainty set in (6) is given as the affine image of a U bounded set = ζ RN , and is given Z { }⊂ Z either (i) by a system of linear inequality constraints P ζ p ≤ or (ii) by a system of Conic Quadratic inequalities P ζ p qT ζ r , i = 1,...,M, k i − i k2≤ i − i or 3 (iii) by a system of Linear Matrix Inequalities dim ζ P + ζ P 0. 0 i i Xi=1 In the cases (ii), (iii) assume also that the system of constraints defining is strictly feasible. U Then the robust counterpart (5) of the uncertain LP (6) is equivalent to – a Linear Programming problem in case (i), – a Conic Quadratic problem in case (ii), – a Semidefinite program in case (iii). In all cases, the data of the resulting robust counterpart problem are readily given by m, n and the data specifying the uncertainty set. Moreover, the sizes of the resulting problem are polynomial in the size of the data specifying the uncertainty set. 2 Proof. A point y = (t,x) Rn+1 is a feasible solution of (5) if and only if y solves a semi-infinite ∈ system of inequalities of the form (C ) [B ζ + β ]T y +[cT ζ + d ] 0 ζ , i i i i i ≥ ∀ ∈ Z (7) i = 0,...,m The fact that y solves (Ci) means that the optimal value in the problem T T min [Biζ + βi] y +[ci ζ + di]: ζ (Pi[y]) ζ ∈ Z n o is nonnegative. Now, by assumption is representable as Z = ζ Rζ r K Z { | − ∈ } where K is either a nonnegative orthant, or a direct product of the second-order cones, or the semidefinite cone.
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